Browse Questions

# The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinate of the point is

$(A)\;an\; ellipse \quad (B)\;parabola \quad (C)\;circle \quad (D)\;rectangular\; hyperbola$

Toolbox:
• Slope of a tangent is $\large\frac{dy}{dx}$
• If the given linear differential equation is of the form $\large\frac{dy}{dx}$$=f(x), then it can be solved by seperating the variables and then integrating it. It is given that the slope of the tangent is equal to the ratio of the abscissa to the ordinate of the point. We know slope of a tangent is \large\frac{dy}{dx} Hence \large\frac{dy}{dx}=\large\frac{x}{y} Now seperating the variables, ydy=xdx On integrating we get \int ydy=\int xdx \large\frac{y^2}{2}=\frac{x^2}{2}$$+c$
Mutiplying throughout by 2
$y^2=x^2+2c=>x^2-y^2+2c=0$
$=>x^2-y^2=k\quad(where \;-2c=k)$
This is an equation of a hyperbola.
Hence the correct option is $D$